Musical Tone Law Method for the Structural Damage Detection

Damage detection tests of inclined cables, steel pipes, spherical shells, and an actual cable-stayed bridge were conducted based on the proposedmusical tone lawmethod.The results show that themusical tone lawmethod could be used in the damage detection of isotropic material structures with simple shape, like cables, pipes, plates, and shells. Having distinct spectral lines like a comb with a certain interval distribution rule is the main characteristic of the music tone law. Damage detection baseline could be established by quantizing the fitting relationship between modal orders and the corresponding frequency values. The main advantage of this method is that it could be used in the structural damage detectionwithout vibration information of an intact structure as a reference.


Introduction
Structural damage detection is an ancient but novel technology.In the past, visual inspections were used to detect the structural damage by trained engineers.However, this traditional method may be costly or inefficient [1].Since Vandiver [2] studied the failure of a steel lighthouse by the change of structural dynamic frequency in 1975, damage detection methods using modal properties such as natural frequencies and modal shapes, which can be obtained from sensors placed on the structure, have been widely discussed in order to assess structural integrity.
Various studies have since then been proposed to promote the development of this field.In general, the past studies have three main characteristics.First, algorithm about the signal processing was mainly studied, such as neural network algorithm, genetic algorithm, Hilbert-Huang algorithm, and EMA method [3][4][5][6].Second, device for the damage detection was developed, such as fiber optic sensors and wireless sensors [7,8].In addition, the application in major engineering was also conducted in recent years [9,10].
There is no doubt that the structure is damaged when structural stiffness decreased obviously.However, the square of frequency is in proportional to stiffness, so when the structure was damaged with attenuation of stiffness, the variation of frequency is not obvious, which has been proved by many research works [11][12][13][14][15].And a baseline is also needed to judge whether the frequency decreased or not.Then, FEA model or test result of intact structures was usually used as the healthy benchmark.For the reason of reliability and measuring accuracy, many difficulties were faced in the actual application.Finding an effective way to detect structural damage by further analysis of frequency information would have a meaningful theoretical and applicable value.
Based on the musical tone law method, three typical structures, including inclined cables with transverse vibration, steel pipes with longitudinal vibration, and spherical shells with out-of-plane vibration, were conducted for the vibration test and damage detection in this paper.The frequencies distributions of intact structures and structures with different damaged levels were collected and analyzed.And a convenient and effective criterion for damage detection is established.Furthermore, the application of this method was tested in an actual cable-stayed bridge.Before going into detail regarding the application project, some basic considerations about the musical tone law method are introduced below.

Musical Tone Law Method
Sound wave is the transmission of vibration in the air.Music tone is the combination of fundamental wave and harmonic wave, and the ratio between frequency of harmonic wave and that of the fundamental wave is a group of regular constants.
The Fourier spectrum of a violin string sound is shown in Figure 1.Note that frequencies of the 1st, 2nd, 4th, and 8th order appear as multiplication distribution, and it is in accord with the music tone law.In auditory sense, the sound meeting the above law is melodious; otherwise, it is always harsh.
Having distinct spectral lines like a comb with a certain interval distribution rule is the main characteristic of the music tone law.
In the past, the quality test of ceramic plates was essentially using the music tone law.Knock test of intact and damaged ceramic plate was also conducted.Three ceramic plates with the same batch were chosen as the test specimens.Specimen 1 and specimen 2 were intact; specimen 3 has a slight crack in the radial direction, as shown in Figure 2. The time history and Fourier spectrum of the knock vibration of specimens 1, 2, and 3 were shown in Figure 3.There is a large difference between the sound signals of the intact and damaged plates.The duration of the intact plate sound is about 0.6 s, while that of the damaged one is only about 0.1 s.The frequency spectrum of intact plate tap is similar to that of musical tone with distinct and regular spectral lines, while the damaged plate tap has indistinct spectral lines.Besides, for the intact plates, the relationship between the spectral lines order and the frequency value remained unchanged to the tap location.For the damaged plate, the relationship changed with different tap locations, but the main feature remained unchanged.
It may propose a new method to explore the change of structural frequency after damage.The intact structural distribution of frequency could serve as a benchmark for damage detection.In the mechanical principle, ceramic plate is similar to a shell structure and both of them follow a similar vibration criterion.The study also found that the frequency lines of one-and two-dimensional elastic structures with regular shape are clear and sharp and have a certain interval distribution.And this phenomenon, named the musical tone law method, could be used in the structural damage detection.Based on this cognition, three types of model structures and an actual engineering were tested in this paper, and the baseline and application range were studied.

Damage Detection Tests of Inclined Cables
Cable-stayed and suspension bridges are two important structural systems of bridges, where the steel cable is one of the most important bearing members.If corrosion or abrasion occurs in the cables, the safety of the whole bridge would be threatened.The following test would study the application of the musical tow law method in the cables.LC0405T piezoelectric accelerometers, NEXUS2692-OS4 charge amplifier, and SigLab20-42 data acquisition instrument were used in all the vibration tests of this paper.And the resolution was set as 0.0625 Hz.
Inclined cables were tightened in the lab to simulate the actual force status, and two test cases, that is, intact and damaged cables, were tested.The cables were formed by one core fiber and 6 cover wires with total length of 28 m, diameter of 10 mm, and unit mass of 0.385 kg/m.The damaged cable was set by cutting 3 cover wires in the midspan with cut length of 2.8 m, as shown in Figure 4.The ends of the cable were fixed, respectively, in the top of the reaction wall and floor, as shown in Figure 5. Tandem tension spring was used to monitor the tension force of the cable, as shown in Figure 6.Overview of the test model is shown in Figure 7.The equivalent sectional area of the damaged cable is 95% of the intact one.As the tension force of the intact cable is 4.5 kN, the tension force of the damaged one was set as 4.25 kN.
Pulse excitation was conducted by knocking the cable by exciting hammer.The accelerometer was also installed in the 1/3 part away from the root of the cable to monitor the transverse vibration, as shown in Figure 8.The stiffness of the tandem tension spring was standardized before the test.Five times of the vibration signals with duration of 40 s were collected, and the transformed autopower spectrum was analyzed.
The theoretical solution of cable frequency is a problem of nonlinear vibration; many methods were studied to solve this problem [16][17][18][19], such as fitting method and finite difference method.Ran and Li [20] used the singular perturbation method to solve the nonlinear dynamic equation and verified this method by tests and numerical simulation.The suggested analytical expression of inclined cable fixed at both ends was developed as in which  denotes the modal orders,  is the average of cable force,  is mass per unite length of the cable, EI is the flexural stiffness of the cable,  is cable length, Δ is the correction term, and  is cable sag.And the 3nd and 5th terms in (1) are corrections of boundary conditions.The expression of inclined cable hinged at both ends could be developed as For flexible cables with short length, the last two terms in (2) could be ignored as an approximate method.Then, the frequency could be solved by (3) [21], where  is the cable force, L is the length of cable, and m is mass per unite length of the cable.However, as the sag effect is significant for the       long cable, the 2nd term in (2) could not be ignored when used for long cables.
The test result is shown in Table 1.The frequencies of the inclined cable vibration mainly appear as multiplication distribution, which suggests that the reliability of (3) based on approximate method is basically satisfactory.For vibration tests of actual long inclined cables, the frequencies could not be fitted as multiplication distribution, and the study in Section 6 shows that the frequencies could be approximately regarded as in exponential distribution.Autopower spectrums of the cable vibration are shown as in Figures 9  and 10.The results show that the intact cable has distinct and regular spectral lines.While the spectral lines interval of the damaged cable changed, the coupling energy around the predominant frequency increased.Figure 11 shows the relationship between the neighbouring frequency interval and the modal orders.For the damaged cable, the neighbouring frequency intervals are rather changeable and no longer equal.Note that the frequencies of the damaged cable are larger than those of the intact one, which do not agree with traditional cases, in which the damaged structure has a lower stiffness.This phenomenon could be explained by (3).As the damaged cable was set by cutting 3 cover wire ropes in the midspan with cut length of 2.8 m, the mass per unite length is lighter than that of the intact one.Thus, the damaged cable has a larger frequency.And the same situation would be met in the next section, that is, tests of steel pipes.

Damage Detection Tests of Steel Pipes
Damage detection tests of one-dimensional steel pipes were conducted, and the longitudinal vibration excited by knock at the pipe end was collected.The test models were hollow straight pipes with length of 9500 mm, external diameter of 61.2 mm, and inner diameter of 51.2 mm, as shown in Figure 12(a).Specimen A was intact, while in the damaged model, specimen B was set by cutting part of the intact pipe in the midspan with cut length of 100 mm and depth of 30 mm, as shown in Figure 12(b).Knock excitation at the pipe end was conducted by exciting hammer, and a piezoelectric accelerometer was installed on the surface of the pipe to monitor the longitudinal vibration, as shown in Figure 12(c).Frequency domain within 2 kHz was analyzed.
The longitudinal vibration frequencies of steel pipes could be solved by ( 4), and the transverse vibration frequency could be solved by (5) based on classical mechanics [22,23].
where  is the order of mode, EI is the flexural stiffness,  is the wave velocity, and  is the sectional area, and this could be solved as  = √/, where E is the elastic modulus of steel (2.1 × 10 5 N/mm 2 ) and  is the steel density (7850 kg/m 3 ).Note that (5)   could be ignored, while the bending stiffness influences are significant for transverse vibration of pipes.In this section, the longitudinal vibration test was performed.Then, the theoretical fundamental frequency of the intact steel pipe could be solved as 272 Hz.The inherent frequencies appear as multiplication distribution.
Autopower spectrums of the pipe vibration are shown as in Figures 13 and 14.Spectral lines of the intact steel pipe (Figure 13) are distinct and the intervals are regular, following the musical tone law.While the spectral lines of the damaged pipe become irregular, some frequencies are even hard to be identified.In this paper, as an intact pipe could be used as a reference, we select the frequency peak of the damaged pipe around the frequency peak of the intact pipe.Besides, the amplitude decreased exponentially compared with the intact pipes.
The above test results show that the musical tone law method could detect the structural damage effectively.The top seven-order frequencies are shown in Table 2. Figure 15 shows the relationship between the neighbouring frequency intervals and the modal order.Spectral lines of the intact pipe followed the theoretical equal-interval distribution, while those of the damaged pipe are rather changeable.So, the structural damage could be detected by quantifying the

Damage Detection Tests of Steel Shells
The application of shell structures increases in recent years, and continuous elastic shell is always adopted in the dynamic analysis, and rigid joint, small deformation, and elastic assumption are also accepted [24].So, the musical tone law method may be used in the damage detection of large-scale shell structures.And small-scale shell model was tested in this section.
Test models are two spherical steel shells with radius of 1000 mm, thickness of 4 mm, and rise of 250 mm.Four accelerometers were installed symmetrically on the edge of the shell, as shown in Figure 16(a).Figure 16(b) shows the test  instruments, which were the same as the above tests.Radial kerf was set artificially in one shell to simulate the damage case, as shown in Figure 17, and the size of the kerf is 1 mm × 40 mm.Two weights of 5 kg were tied on both ends of the shell along the diameter by strings, which aimed to set an initial deformation.Then, free vibration was excited by burning out the strings suddenly.Figure 18 shows the spectral lines of both shells.The measured frequencies are shown in Table 3.Note that frequencies of the intact shell no longer follow the equalinterval distribution.Exponential fitting of the measured frequencies was conducted; the result is shown in Figure 19.Goodness of fit index  2 was used to evaluate the fitting and it is calculated as (6).The value is between 0 and 1.As the value of  2 is closer to 1, the fitting goodness is more satisfactory.In (6),   is the actual value;   is the value in the fitting curve.Equations ( 7) and ( 8) are the fitting function of the intact and damaged shells, in which  denotes the modal order and  denotes the frequency value.
The results show that the special lines of the intact shell are distinct, while multimodal phenomenon around the predominant frequency is quite obvious in the damaged shell.Actually, the frequency peak of the damaged shell is hard to be selected.In this paper, as an intact shell could be used as a reference, we select the frequency peak of the damaged shell around the frequency peak of the intact shell.Although the fitted  2 of intact and damaged shells are closer as 99.93% and 99.62%, the spectral lines of the damaged one are multimodal, not in accord with the comblike distribution, which is similar to the ceramic plate test in Figure 3. So, the radial damage could be detected based on the musical tone law method.And during the test, the hammering sound of the intact shell was clangorous and perpetual, while that of the damaged shell was depressing and rushing.This phenomenon agreed with the knock test of ceramic plates, in which the musical tone law method was proposed.

Baseline of the Musical Tone Law Method and Engineering Application
In the above tests, vibration of small-scale inclined cable and steel pipe is one-dimensional and that of steel shell is two-dimensional.The results show that frequencies of smallscale inclined cable and steel pipes appear as multiplication distribution, and the interval between neighbouring Advances in Materials Science and Engineering  frequencies is equal.Frequencies of two-dimensional shells appear as exponential distribution.In the elastic mechanics, theoretical frequencies of cantilever also present as equalinterval distribution.In conclusion, the musical tone law method could be used in the damage detection of isotropic material structures with simple shape, like cables, pipes, plates, and shells.In general, frequencies of one-dimensional vibration structures present as multiplication distribution, while those of two-dimensional vibration structures present as exponential or logarithm distribution.So, the goodness of fit could be used as a baseline in the damage detection.
The application of the above method was examined in an actual cable-stayed bridge.Theoretical solution of long cable frequencies shows that the relationship between the modal order and the frequency value is exponential distribution.The tested Songyuan bridge is a cable-stayed bridge with total length of 2546.5 m.Two symmetrical bridge towers were built in the midspan.The overview of the bridge is shown in Figure 20.Vibration tests of two symmetrical cables at either side of the tower were conducted, and test results of the two cables are almost the same.The test instruments including data acquisition, charge amplifier, and piezoelectric accelerometer are all same as the above model test (Figure 21).The sampling bandwidth was set as 200 Hz, the resolution was set as 0.0156 Hz, and the sampling duration was set as 200 s.Fourier spectrum of time history signal is shown in Figure 22.Table 4 shows the top ten frequencies of the cable.Fitting curve of the relationship between the modal order and the The test and fitting results show that the spectral lines are distinct and like a comb, frequency presented as exponential distribution, and the goodness of fit is more than 99.9%.The example shows that the baseline of the musical tone law method could be used in the damage detection of actual engineering.

Conclusions
Based on the phenomenon of musical tone law, damage detection tests of three simple models and an actual cablestayed bridge were conducted.The following conclusions were obtained by the test results and further analysis.
(1) The musical tone law method could be used in the damage detection of isotropic material structures with simple shape, like cables, pipes, plates, and shells.The frequency lines of the above intact structures are distinct and like a comb.And the multimodal phenomenon around the predominant frequency is quite obvious in the damaged structures.
(2) In general, frequencies of one-dimensional vibration structures, like pipes, present as multiplication distribution, while those of two-dimensional vibration structures, like shells and long cables, present as exponential or logarithm distribution.So, the distribution characteristic of structural frequencies and the fitting goodness could be used as a baseline in the damage detection.And the damage level could be quantized by the value of fitting goodness in the congeneric structures.
(3) The main advantage of this method is that it could be used in the structural damage detection without vibration information of an intact structure as a reference.And the application of the above method was examined in an actual cable-stayed bridge.

Figure 1 :
Figure 1: Fourier spectrum of a violin string sound.

Figure 2 :
Figure 2: Three ceramic plates for the test.

Figure 3 :
Figure 3: Time history and FFT of the plates tap.

Figure 4 :
Figure 4: Damaged part of the cable.

Frequency 20 Figure 10 :
Figure 10: Spectral lines of the damaged cable.

Figure 11 :
Figure 11: Neighbouring frequency interval of both models.

Figure 16 :KerfFigure 17 :
Figure 16: Specimen and test setup for damage detection of steel shells.

Figure 18 :
Figure 18: Spectral lines of the intact and damaged shells.

Figure 19 :
Figure 19: Comparison of spectrum fitting of intact and damaged shells.

Table 1 :
Frequencies of the inclined cables.

Table 2 :
Natural frequencies of steel pipes under various damage cases.

Table 3 :
Natural frequencies of both shells.

Table 4 :
Tested frequencies of the cable.